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• CommentRowNumber1.
• CommentAuthorColin Tan
• CommentTimeJul 19th 2014
• (edited Jul 19th 2014)

At holomorphic function, a function between complex manifolds is defined as holomorphic if it is complex-differentiable. Given that underlying a complex manifold is a smooth manifold, ought we use the smooth structure to define a holomorphic function between complex manifolds as a smooth function annihilated by the antiholomorphic differential?

From my understanding, complex-differentiability is pertient when we speak of functions, say, from ${\mathbb{C}}$ to ${\mathbb{C}}$. There we may take ${\mathbb{C}}$ as a field without cohesion, which is enough to define complex differentiability. That is what I understand to be the point about Goursat theorem v.s. Cauchy’s original result. After this hard analysis is done, then lifting the definition of holomorphicity to complex manifolds could be done using the antiholomorphic differential.

Referring to Urs’ work on cohesion in complex geometry, the subleties of defining “holomorphic” as “complex-differentiability” could arise when studying functions between more general complex analytic space, as opposed to the traditional setting of complex manifolds.

• CommentRowNumber2.
• CommentAuthorColin Tan
• CommentTimeJul 19th 2014

To add to the previous comment, perhpas we can tighten the relation between Doubeault complex and holomorphic function with the joint concept of “antiholomorphic differential”.

Browsing analytic geometry, under the section “Holomorphic functions in several complex variables”, a link named “holomorphic function” actually points to the entry “Doubeault complex” instead.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 19th 2014

currently Dolbeault operator redirects to Dolbeault complex.

Please feel invited to split it off as a separate entry and to make “antiholomorphic differential” either the title of or a redirect to that page.

• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeJul 20th 2014

I do not know of any situations where a function is complex-differentiable without being smooth, but if there are any, then I would be inclined to use “holomorphic” only for those that are smooth (and annihilated by $\bar{\partial}$). In other words, I agree.

• CommentRowNumber5.
• CommentAuthorColin Tan
• CommentTimeJul 20th 2014
To push the definition to the limit, complex differentiability could be defined for algebras over the complex numbers. Without some topology, say a compatible Banach space structure, it would likely be that such complex differentiable functions are not smooth.

Perhaps we should distinguish between holomorphic and complex differentability. Perhaps as you say, holomorphic ought to be "smooth and annihalted by the antiholomorphic differential".
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